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Weak Identification with Bounds in a Class of Minimum Distance Models

Gregory Fletcher Cox

Abstract

When parameters are weakly identified, bounds on the parameters may provide a valuable source of information. Existing weak identification estimation and inference results are unable to combine weak identification with bounds. Within a class of minimum distance models, this paper proposes identification-robust inference that incorporates information from bounds when parameters are weakly identified. This paper demonstrates the value of the bounds and identification-robust inference in a simple latent factor model and a simple GARCH model. This paper also demonstrates the identification-robust inference in an empirical application, a factor model for parental investments in children.

Weak Identification with Bounds in a Class of Minimum Distance Models

Abstract

When parameters are weakly identified, bounds on the parameters may provide a valuable source of information. Existing weak identification estimation and inference results are unable to combine weak identification with bounds. Within a class of minimum distance models, this paper proposes identification-robust inference that incorporates information from bounds when parameters are weakly identified. This paper demonstrates the value of the bounds and identification-robust inference in a simple latent factor model and a simple GARCH model. This paper also demonstrates the identification-robust inference in an empirical application, a factor model for parental investments in children.

Paper Structure

This paper contains 14 sections, 8 theorems, 32 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. A Class of Minimum Distance Models
  3. Estimation Limit Theory
  4. Consistency
  5. Identification Strength
  6. Convergence Rate
  7. The Influence of the Boundary
  8. Limit Theory
  9. Identification-Robust Inference
  10. Quasi-Likelihood Ratio Statistic
  11. Inference Recommendation
  12. Simulations
  13. Empirical Application
  14. Conclusion

Key Result

Lemma 1

Under Assumptions 1, 2, 3, and 4, there exists a sequence of neighborhoods of $\pi_\ast$, $\Pi_n$, such that $\mathcal{B}_n:=\mathcal{B}(\Pi_n)$ is compact and satisfies

Figures (2)

  • Figure 1: Simulated Rejection Probabilities of Tests in Example 1: $b=0$
  • Figure 2: Simulated Power Functions of Tests in Example 1: $b>0$

Theorems & Definitions (16)

  • Example : 1: Factor Model
  • Example : 2: GARCH Model
  • Lemma 1
  • Example : 1, Continued
  • Example : 2, Continued
  • Example : 1, Continued
  • Example : 2, Continued
  • Lemma 2
  • Lemma 3
  • Lemma 4
  • ...and 6 more